The homology groups of a compact, oriented, connected, n-dimensional manifoldX have a fundamental property called Poincaré duality: there is a perfect pairing Classically—going back, for instance, to Henri Poincaré—this duality was understood in terms of intersection theory. An element of is represented by a j-dimensional cycle. If an i-dimensional and an -dimensional cycle are in general position, then their intersection is a finite collection of points. Using the orientation of X one may assign to each of these points a sign; in other words intersection yields a 0-dimensional cycle. One may prove that the homology class of this cycle depends only on the homology classes of the original i- and -dimensional cycles; one may furthermore prove that this pairing is perfect. When X has singularities—that is, when the space has places that do not look like —these ideas break down. For example, it is no longer possible to make sense of the notion of "general position" for cycles. Goresky and MacPherson introduced a class of "allowable" cycles for which general position does make sense. They introduced an equivalence relation for allowable cycles, and called the group of i-dimensional allowable cycles modulo this equivalence relation "intersection homology". They furthermore showed that the intersection of an i- and an -dimensional allowable cycle gives an zero-cycle whose homology class is well-defined.
Stratifications
Intersection homology was originally defined on suitable spaces with a stratification, though the groups often turn out to be independent of the choice of stratification. There are many different definitions of stratified spaces. A convenient one for intersection homology is an n -dimensional topological pseudomanifold. This is a spaceX that has a filtration of X by closed subspaces such that:
for each i and for each point x of Xi − Xi−1, there exists a neighborhood of x in X, a compact -dimensional stratified spaceL, and a filtration-preserving homeomorphism. Here is the open cone on L.
Xn−1 = Xn−2
X − Xn−1 is dense in X.
If X is a topological pseudomanifold, the i-dimensional stratum of X is the space Xi − Xi−1. Examples:
If X is an n-dimensional simplicial complex such that every simplex is contained in an n-simplex and n−1 simplex is contained in exactly two n-simplexes, then the underlying space of X is a topological pseudomanifold.
If X is any complex quasi-projective variety then its underlying space is a topological pseudomanifold, with all strata of even dimension.
Perversities
Intersection homology groups IpHi depend on a choice of perversity p, which measures how far cycles are allowed to deviate from transversality. A perversityp is a function from integers ≥2 to integers such that
p = 0
p − p is 0 or 1
The second condition is used to show invariance of intersection homology groups under change of stratification. The complementary perversityq of p is the one with Intersection homology groups of complementary dimension and complementary perversity are dually paired. Examples:
The minimal perversity has p = 0. Its complement is the maximal perversity with q = k − 2.
The middle perversitym is defined by m = integer part of /2. Its complement is the upper middle perversity, with values the integer part of /2. If the perversity is not specified, then one usually means the lower middle perversity. If a space can be stratified with all strata of even dimension then the intersection homology groups are independent of the values of the perversity on odd integers, so the upper and lower middle perversities are equivalent.
Singular intersection homology
Fix a topological pseudomanifold X of dimension n with some stratification, and a perversity p. A map σ from the standard i-simplex Δi to X is called allowable if is contained in the i − k + p skeleton of Δi. The complex Ip is a subcomplex of the complex of singular chains on X that consists of all singular chains such that both the chain and its boundary are linear combinations of allowable singular simplexes. The singular intersection homology groups are the homology groups of this complex. If X has a triangulation compatible with the stratification, then simplicial intersection homology groups can be defined in a similar way, and are naturally isomorphic to the singular intersection homology groups. The intersection homology groups are independent of the choice of stratification of X. If X is a topological manifold, then the intersection homology groups are the same as the usual homology groups.
Small resolutions
A resolution of singularities of a complex varietyY is called a small resolution if for every r > 0, the space of points of Y where the fiber has dimension r is of codimension greater than 2r. Roughly speaking, this means that most fibers are small. In this case the morphism induces an isomorphism from the homology of X to the intersection homology of Y. There is a variety with two different small resolutions that have different ring structures on their cohomology, showing that there is in general no natural ring structure on intersection homology.
Sheaf theory
Deligne's formula for intersection cohomology states that where ICp is a certain complex of sheaves on X. The complex ICp is given by starting with the constant sheaf on the open set X−Xn−2 and repeatedly extending it to larger open setsX−Xn−k and then truncating it in the derived category; more precisely it is given by Deligne's formula where τ≤p is a truncation functor in the derived category, ik is the inclusion of X−Xn−k into X−Xn−k−1, and is the constant sheaf on X−Xn−2. By replacing the constant sheaf on X−Xn−2 with a local system, one can use Deligne's formula to define intersection cohomology with coefficients in a local system.
Properties of the complex IC(''X'')
The complex ICp has the following properties
On the complement of some closed set of codimension 2, we have
is 0 for i + m < 0
If i > 0 then is zero except on a set of codimension at least a for the smallest a with p ≥ m − i
If i > 0 then is zero except on a set of codimension at least a for the smallest a with q ≥
As usual, q is the complementary perversity to p. Moreover, the complex is uniquely characterized by these conditions, up to isomorphism in the derived category. The conditions do not depend on the choice of stratification, so this shows that intersection cohomology does not depend on the choice of stratification either. Verdier duality takes ICp to ICq shifted by n = dim in the derived category.
